Appl. Gen. Topol. 15, no. 1 (2014), 85-92 @ doi:10.4995/agt.2014.1897 c AGT, UPV, 2014

Baire spaces and hyperspace revisited

Steven Bourquin and Laszl´ o´ Zsilinszky

Department of Mathematics and Computer Science, The University of North Carolina at Pem- broke, Pembroke, NC 28372, USA ([email protected], [email protected])

Abstract

It is the purpose of this paper to show how to use approach spaces to get a unified method of proving Baireness of various hyperspace topologies. This generalizes results spread in the literature including the general (proximal) hit-and-miss topologies, as well as various topologies gener- ated by gap and excess functionals. It is also shown that the Vietoris hyperspace can be non-Baire even if the is a 2nd countable Hausdorff .

2010 MSC: Primary 54B20; Secondary 54E52, 54A05 Keywords: Baire spaces, approach spaces, (proximal) hit-and-miss topolo- gies, weak hypertopologies, Banach-Mazur game

1. Introduction A is a Baire space [7] provided countable collections of dense open subsets have a dense intersection. Baireness is one of the weakest completeness properties, and yet, it has fundamental applications throughout mathematics. This is why there has been interest in investigating Baireness, along with other completeness properties, in the theory of hyperspace topolo- gies which in turn have applications in various branches of mathematics on their own ([1], [12]). In the present paper we will continue in this research by exhibit- ing some common features of the plethora of studied hyperspaces as far as their Baireness is concerned. Indeed, following the unified exposition of hyperspace topologies introduced in [20], we prove Baireness of these general hyperspaces in

Received 16 November 2013 – Accepted 13 February 2014 S. Bourquin and L. Zsilinszky one theorem, thereby generalizing and extending several results about specific hyperspaces spread in the literature ([1],[14],[19],[23],[21],[3],[4],[9]). In hyperspace theory, it is customary to assume that the base space is Haus- dorff (or at least T1), since then singletons are closed, and the base space embeds into the hyperspace. It was observed that imposing separation axioms on the base space is frequently not necessary to obtain results on hypertopologies (see [6], [19], [24], [10]), which is the case throughout this paper as well. At the end, an example is provided that shows the limitations of Baireness results for the Vietoris .

2. Preliminaries Throughout the paper ω stands for the non-negative integers, P(X) for the power , CL(X) for the nonempty closed subsets of a topological space X, and Ec for the complement of E ⊂ X in X. The general description of the hyperspaces we will use was given in [20], here we just provide the definitions so the paper is self-contained, more detail can be found in [20], and the references therein. Suppose that (X,δ) is an approach space [13], i.e. X is a nonempty set and δ : X ×P(X) → [0, ∞] is a so-called distance (on X) having the following properties: (D1) ∀x ∈ X : δ(x, {x})=0 (D2) ∀x ∈ X : δ(x, ∅)= ∞ (D3) ∀x ∈ X ∀A, B ⊂ X : δ(x, A ∪ B) = min{δ(x, A),δ(x, B)}

(D4) ∀x ∈ X ∀A ⊂ X ∀ε> 0 : δ(x, A) ≤ δ(x, Bε(A)) + ε, where Bε(A) = {x ∈ X : δ(x, A) ≤ ε}; we will also use the notation Sε(A) = {x ∈ X : δ(x, A) <ε}. Every approach space (X,δ) generates a topology τδ on X defined by the closure operator: A¯ = {x ∈ X : δ(x, A)=0}, A ⊂ X. The functional D : P(X) ×P(X) → [0, ∞] will be called a gap provided: (G1) ∀A, B ⊂ X : D(A, B) ≤ inf δ(a,B) a∈A (G2) ∀x ∈ X ∀A ⊂ X : D({x}, A) = inf δ(y, A) y∈{x} (G3) ∀A, B, C ⊂ X : D(A ∪ B, C) = min{D(A, C),D(B, C)}. In the sequel (unless otherwise stated) X will stand for an approach space (X,δ) with a gap D (denoted also as (X,δ,D)). Remark 2.1. (1) Examples of distances: • [13] Let X be a topological space. For x ∈ X and A ⊂ X define

0, if x ∈ A,¯ δt(x, A)= (∞, if x∈ / A.¯

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Then δt is a distance on X, and τδt coincides with the topology of X. • Let (X, U) be a . Then U is generated by the family D of uniform pseudo-metrics on X such that d ≤ 1 for all d ∈ D, and d1, d2 ∈ D implies max{d1, d2} ∈ D. Then

δu(x, A) = sup d(x, A), x ∈ X, A ⊂ X d∈D

defines a (bounded) distance on X and τδu coincides with the topology induced by U on X. • Let (X, d) be a space. Then δm(x, A)= d(x, A) is a distance

on X and τδm is the topology generated by d on X. (2) Examples of gaps: • For an arbitrary approach space (X,δ) D(A, B) = inf δ(a,B) a∈A is clearly a gap on X. This is how we are going to define the gap Dt (resp. Dm) in topological (metric) spaces using the relevant distance δt (resp. δm). • Let (X, U) be a uniform space generated by the family D of uni- form pseudo-metrics on X bounded above by 1. For A, B ⊂ X define

Du(A, B) = sup inf d(a,B). d∈D a∈A

Then Du is a gap on X. (3) Excess functional: for all A, B ⊂ X define the excess of A over B by e(A, B) = sup δ(a,B). a∈A

The symbols et,eu,em will stand for the excess in topological, uniform and metric spaces, respectively defined via δt,δu,δm.

3. Hyperspace topologies For E ⊂ X write E− = {A ∈ CL(X): A ∩ E 6= ∅}, E+ = {A ∈ CL(X): A ⊂ E} and E++ = {A ∈ CL(X): D(A, Ec) > 0}. In what follows ∆1 ⊂ CL(X) is arbitrary and ∆2 ⊂ CL(X) is such that

∀ε> 0 ∀A ∈ ∆2 ⇒ Sε(A) is open in X. D D ∅ k+1 k+1 2k+2 Denote = (∆1, ∆2)= k∈ω(∆1 ∪{ }) × (∆2 ∪{X}) × (0, ∞) . Whenever referring to some S,T ∈ D, we will assume that for some k,l ∈ ω S S = (S0,...,Sk; S˜0,..., S˜k; ε0,...,εk;˜ε0,..., ε˜k)

T = (T0,...,Tl; T˜0,..., T˜l; η0,...,ηl;˜η0,..., η˜l).

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For S ∈ D denote c ˜ M(S)= (Bεi (Si)) ∩ Sε˜i (Si) and i k \≤ ∗ S = {A ∈ CL(X): D(A, Si) >εi and e(A, S˜i) < ε˜i}. i k \≤ For U0,...,Un ∈ τδ \{∅} and S ∈ D denote − ∗ (U0,...,Un)S = Ui ∩ S , i n \≤ [U0,...,Un]S = (Ui ∩ M(S)) × M(S). i n i>n Y≤ Y It is easy to see that the collections ∗ B = {(U0,...,Un)S : U0,...,Un ∈ τδ \{∅},S ∈ D,n ∈ ω},

B = {[U0,...,Un]S : U0,...,Un ∈ τδ \{∅},S ∈ D,n ∈ ω} form a base for topologies on CL(X) and Xω, respectively; denote them by τ ∗, and τ, respectively. Remark 3.1. Note that τ is a “pinched-cube” topology as defined in [3], [23]; indeed, we just need to take ∆ = {M(S)c : S ∈ D}. Remark 3.2.

• Let (X, τ) be a topological space. Let ∆1 = ∆ and ∆2 = {X}. Then c + for B ∈ ∆ and ε,η > 0, {A ∈ CL(X): Dt(A, B) > ε} = (B ) and ∗ + {A ∈ CL(X): et(A, X) < η} = CL(X). Thus τ = τ is the general hit-and-miss topology on CL(X) (see [17], [1], [8], [19], [3]). Choosing ∆ = CL(X) we get the most studied hit-and-miss topology, the so- called Vietoris topology τV (cf. [15], [5], [1]); a typical base element for τV is

(U0,...,Un)= {A ∈ CL(X): A ⊆ Ui and A ∩ Ui 6= ∅ for all i ≤ n}. i n [≤ • Let (X, U) be a uniform space. Let ∆1 = ∆ and ∆2 = {X}. Then c ++ for B ∈ ∆ and ε,η > 0, {A ∈ CL(X): Du(A, B) > ε} = (B ) ∗ ++ and {A ∈ CL(X): eu(A, X) < η} = CL(X). Thus τ = τ is the proximal hit-and-miss topology on CL(X) (see [1], [19], [3]). • Let (X, d) be a . Let ∆1, ∆2 ⊂ CL(X) be such that ∆1 contains the singletons. Then τ ∗ coincides with the weak hypertopology τweak generated by gap and excess functionals (see [2], [9], [20]). As we indicated in the Introduction, we do not assume any separation axiom on X, however, the following property seems to be necessary: we will say that (X,δ,D) has property (P) provided

∀x ∈ X ∀k ∈ ω ∀A0,...,Ak ∈ CL(X) ∀ε0,...,εk > 0 ∃y ∈ {x} :

δ(x, Ai) >εi =⇒ D({y}, Ai) >εi for all i ≤ k.

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This property is satisfied in uniform and metric spaces, and a topological space X has property (P) iff X is weakly-R0 [24], i.e. for all open U ⊆ X and x ∈ U there is a y ∈ {x} with {y} ⊂ U iff each nonempty difference of open sets contains a nonempty . We will say that the family D is weakly quasi-Urysohn [20] provided for all ∗ ∅ 6= (U0,...,Un)S ∈ B there is a T ∈ D such that ∅ 6= (U0,...,Un)T ⊂ (U0,...,Un)S and (*) ∀E countable : E ⊂ M(T )=⇒ E ∈ S∗. The translation of this property for the (proximal) hit-and-miss topologies is as follows: given a topological (uniform) space X (resp. (X, U)), the family ∆ ⊆ CL(X) is said to be (uniformly) weakly quasi-Urysohn provided whenever S ∈ ∆ is disjoint to some nonempty open Ui ⊆ X (i ≤ n), there exists T ∈ ∆ c such that Ui ∩ T 6= ∅ for all i ≤ n, S ⊂ T , and ∀E countable : (E ⊂ T c =⇒ E ⊂ Sc) (resp. E ⊂ U[S]c for some U ∈U). We will say that X is (weakly) quasi-regular provided every nonempty open U ⊂ X has a nonempty open subset V such that V ⊆ U (resp. E ⊂ U for all countable E ⊂ V ). It is not hard to see that if X is weakly quasi-regular, then CL(X) is weakly quasi-Urysohn. Proposition 3.3 ([20, Lemma 3.1]). labelZs2 Suppose that (X,δ,D) has prop- ∗ erty (P), and (U0,...,Un)S , (V0,...,Vm)T ∈B . Then (U0,...,Un)S ⊂ (V0,...,Vm)T implies M(S) ⊂ M(T ), and for all j ≤ m there exists i ≤ n with M(S) ∩ Ui ⊂ M(T ) ∩ Vj . Although it would be possible to establish Baireness of (CL(X), τ ∗) using the so-called Banach-Mazur game, as done in [14], [19], or [3], we have chosen a different method, which makes the proofs more transparent and actually improves on results of the above mentioned papers. We will need some auxiliary material: if Y,Z are topological spaces, the mapping f : Y → Z is said to be feebly continuous [7] provided intf −1(U) 6= ∅ for each open U ⊂ Z with f −1(U) 6= ∅; further f is δ-open [7] provided f(A) is somewhere dense in Z for every somewhere dense A of Y . Proposition 3.4 ([7, Theorem 4.7]). If f is a feebly continuous δ-open func- tion from a Baire space onto a space Y , then Y is a Baire space. The following theorem generalizes [14, Theorem 3.8], [19, Theorem 4.1], and [3, Theorem 2.5]: Theorem 3.5. Suppose that X has property (P) and D is a weakly quasi- Urysohn family. If (Xω, τ) is a Baire space, then so is (CL(X), τ ∗). Proof. Denote S(X) = {A ∈ CL(X): A is separable}. Then S(X) is dense in (CL(X), τ ∗), since even the set of closures of finite subsets of X is. Thus, ∗ it suffices to prove that (S(X), τ ↾S(X)) is a Baire space. Define the mapping f : Xω → S(X) via ω f((xk)k)= {xk : k ∈ ω}, where (xk)k ∈ X .

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We will show that f is a feebly continuous δ-open function, so Proposition 3.4 will apply. −1 To see feeble continuity, take a U = (U0,...,Un)S∩S(X) such that f (U) 6= ∅. Now if we take the T ∈ D from weak quasi-Urysohnness of D corresponding −1 to (U0,...,Un)S, then (*) virtually claims that ∅ 6= [U0,...,Un]T ⊂ f (U). To justify δ-openness of f, take an A ⊂ Xω which is dense in some ∅ 6= ∗ V = [V0,...,Vm]P ∈ B. Then f(A) is dense in V = (V0,...,Vm)P ∩ S(X). ∗ ∗ Indeed, if U = (U0,...,Un)S ∩ S(X) is a nonempty open subset of V , take the T ∈ D from weak quasi-Urysohnness of D corresponding to (U0,...,Un)S. Then in view of Proposition ??,[U0,...,Un]T is a nonempty open subset of V. Consequently, we can find an (xk)k ∈ A ∩ [U0,...,Un]T . Then E = ∗ {xk : k ∈ ω} ⊂ M(T ), so by (*), f((xk)k) = E ∈ S , and hence f((xk)k) ∈ f(A) ∩ U∗.  A collection P of nonempty open subsets in a space X is a π-base, if every nonempty open subset of X contains at least one member of P; moreover, P is a countable-in-itself π-base [22], provided each member of P contains countably many members of P. Corollary 3.6. Suppose that X is a Baire space with a countable-in-itself π- base and it has property (P). Suppose that D is a weakly quasi-Urysohn family. Then (CL(X), τ ∗) is a Baire space. Proof. Since τ is a pinched-cube topology (see Remark 3.1), it follows by [23, Theorem 2.1], that (Xω, τ) is a Baire space, so Theorem 3.5 applies. 

4. Applications

The following theorem improves [19, Corollary 4.2], and [3, Theorem 4.1,The- orem 5.1]:

Theorem 4.1. Suppose that X is a weakly-R0 (resp. uniform) space having a countable-in-itself π-base, which is also a Baire space. If ∆ ⊂ CL(X) is a (uni- formly) weakly quasi-Urysohn subfamily, then (CL(X), τ +) (resp. (CL(X), τ ++)) is a Baire space. The previous theorem yields the following corollary, which slightly general- izes [14, Corollary 3.9] (see also [4, Corollary 2.2]):

Theorem 4.2. Let (X, τ) be a weakly-R0, weakly quasi-regular Baire space having a countable-in-itself π-base. Then (CL(X), τV ) is a Baire space. In the next example we show that weak quasi-regularity of X is an essential condition in the Baireness results concerning the Vietoris topology. To do this we need to recall some basic facts about the Banach-Mazur game BM(X) (see [7] or [11]) played by players α and β who take turns choosing sets from a fixed π-base P in the topological space X: β starts by picking V0, then α responds by choosing some U0 ⊆ V0. Continuing like this, while each player picks a

c AGT, UPV, 2014 Appl. Gen. Topol. 15, no. 1 90 Baire spaces and hyperspace topologies revisited nonempty contained in the previous choice of the opponent, one gets ∅ a run V0,U0,...,Vn,Un,... of BM(X), which is won by α, if i<ω Ui 6= , otherwise, β wins. The key result about the Banach-Mazur game is that X is not a Baire space iff β has a winning tactic in BM(X), i.e. thereT is a function t : P∪{∅}→P such that if t(∅) = V0, and t(Un) = Vn+1 for each n<ω, then β wins. Example 4.3. There exists a Hausdorff, Baire, 2nd countable space such that (CL(X), τV ) is not Baire. Proof. Let E be the Euclidean topology on R, and B ⊂ R be a Bernstein set [7], i.e. such that both B and R \ B meet every uncountable Gδ subset of R. Define a finer than E topology EB on R as follows:

EB = {U ∪ V ∩ B : U, V ∈ E}.

Then X = (R, EB) is clearly Hausdorff and 2nd countable. In order to prove that (CL(X), τV ) is β-favorable, we will play the Banach- Mazur game using the following π-base for the hyperspace:

PV = {(I0 ∩ B,...,In ∩ B): I0,...,In are pairwise disjoint open intervals}. + Define a tactic tV for β in BM(CL(X)) as follows: put tV (∅)= B ; moreover, if U = (I0 ∩ B,...,In ∩ B) ∈PV , for each i ≤ n, choose disjoint bounded open E 0 1 0 1 intervals Ji ,Ji so that Ji ∪ Ji ⊆ Ii, and put 0 1 0 1 tV (U) = (J0 ∩ B,J0 ∩ B,...,Jn ∩ B,Jn ∩ B). + Let B , U0,tV (U0),..., Uk,tV (Uk),... be a run of BM(CL(X)) compatible with tV . Suppose that there is some A ∈ k Uk. Then A is a EB-closed subset of B, and since the E-, and EB-closure of subsets of B coincide, A would be a E- T closed, and in view of the definition of tV , also dense-in-itself subset of B, which ∅ contradicts the definition of the Bernstein set. Consequently, k Uk = , and β wins the run.  T Remark 4.4. It follows by [18, Example 2.7], that the previous example is not quasi-regular, but it is not even weakly quasi-regular by Theorem 4.2. It is also an example showing that quasi-regularity cannot be removed in the key result of [4, Theorem 2.1], stating that if X is a quasi-regular space such that Xω is a Baire space, then (CL(X), τV ) is a Baire space, since by a result of Oxtoby [16] products of 2nd countable Baire spaces are Baire. Our last result generalizes [9, Theorem 4.2] and [20, Corollary 6.1]:

Theorem 4.5. Let (X, d) be a separable metric Baire space, and ∆1, ∆2 ⊂ CL(X) be such that ∆1 contains the singletons. Then (CL(X), τweak) is a Baire space.

Proof. It follows from Corollary 3.6, since D(∆1, ∆2) is weakly quasi-Urysohn by [20, Remark 3.1(iii)]. 

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